Consider a power system consisting of $N$ number of buses. Buses in this power system are categorized into slack bus. $P V$ buses and $P Q$ buses for load flow study. The number of $P Q$ buses is $N_L$. The balanced Newton-Raphson method is used to carry out load flow study in polar form $H, S, M$ and $R$ are sub-matrices of the Jacobian matrix $J$ as shown below:

$$ \left[\begin{array}{l} \Delta P \\ \Delta Q \end{array}\right]=J\left[\begin{array}{l} \Delta \delta \\ \Delta \gamma \end{array}\right] \text {, where } J=\left[\begin{array}{ll} H & S \\ M & R \end{array}\right] $$

The dimension of the sub matrix $M$ is

A 1000 $$ \times $$ 1000 bus admittance matrix for an electric power system has 8000 non-zero elements. The minimum number of branches (transmission lines and transformers) in this system are _____ (up to 2 decimal places).

A $$3$$-bus power system is shown in the figure below, where the diagonal elements of $$Y$$-bus matrix are: $${Y_{11}} = - j12\,pu,\,\,\,{Y_{22}} = - j15\,pu\,\,$$ and $$\,{Y_{33}} = - j7\,pu.$$

The per unit values of the line reactance's $$p, q$$ and $$r$$ shown in the figure are

A 10-bus power system consists of four generator buses indexed as G1, G2, G3, G4 and six load buses indexed as L1, L2, L3, L4, L5, L6. The generator bus G1 is considered as slack bus, and the load buses L3 and L4 are voltage controlled buses. The generator at bus G2 cannot supply the required reactive power demand, and hence it is operating at its maximum reactive power limit. The number of non-linear equations required for solving the load flow problem using Newton-Raphson method in polar form is ____________.